An orthonormal set in an inner product space is a collection of vectors that are mutually perpendicular and each of unit length. In symbols, a family {u_i} is orthonormal when ⟨u_i,u_j⟩ = 0 for i ≠ j and ⟨u_i,u_i⟩ = 1 for every i. This combines two familiar ideas: orthogonality (zero inner product) and normalization (norm equals one).
Basic properties
Orthonormal systems enjoy simple algebraic and geometric properties that make them central in linear algebra and functional analysis. Important facts include:
- Kronecker delta: ⟨u_i,u_j⟩ = δ_{ij}, where δ_{ij} is 1 when i=j and 0 otherwise.
- Linear independence: any orthonormal set is linearly independent (except the empty set).
- Projection simplicity: projection of x onto a unit vector v is ⟨x,v⟩ v.
- Parseval/Plancherel: for an orthonormal basis, the sum of squared coefficients equals the squared norm of the vector.
Examples and constructions
Standard examples include the standard basis e_i in R^n or C^n, and orthonormal trigonometric functions used in Fourier series. If one starts with a linearly independent family, the Gram–Schmidt process produces an orthonormal family spanning the same subspace; this procedure is named after J. P. Gram and Erhard Schmidt.
Uses and importance
Orthonormal bases simplify calculations: coordinates are given by inner products and matrices representing orthonormal change of basis are orthogonal or unitary, preserving lengths and angles. Applications appear across mathematics and engineering, for example in numerical linear algebra, signal processing, quantum mechanics, and approximation theory.
Distinctions and remarks
Do not confuse an orthogonal set (vectors mutually perpendicular but not necessarily unit length) with an orthonormal set. When a full basis of a space is orthonormal it is called an orthonormal basis, and every vector decomposes uniquely as a sum of basis vectors weighted by inner products. For further context see inner product space and the notion of a unit vector.